Simultaneous approximation of zero by values of integral polynomials with respect to different valuations. (English) Zbl 1108.11050

Kovalevskaya, Ella

Math. Slovaca 54, No. 5, 479-486 (2004).

Summary: We prove an analogue of the convergence part of Khintchine’s theorem for the simultaneous approximation of zero in \(\mathbb R\times \mathbb C\times \mathbb Q_p\) by the values of polynomials \(P_n(y)\in \mathbb Z[y]\). This is a proof of a stronger version of V. Sprindzhuk’s conjecture (1980).

MSC:

11J61	Approximation in non-Archimedean valuations
11J83	Metric theory

Keywords:

Diophantine approximation; Khintchine theorem; integral polynomials; Lebesgue measure; Haar measure

References:

[1]	BERESNEVICH V.-BERNIK V.-KLEINBOCK D.-MARGULIES G. A.: Metric Diophantine approximation: the Khintchine-Groshev theorem for non-degenerate manifolds. Mosc. Math. J. 2 (2002), 203-225. · Zbl 1013.11039
[2]	BERNIK V. I.-BORBAT V. N.: Simultaneous approximation of zero values of integral polynomials in \({\mathbb Q}_p\). Proc. Steklov Inst. Math. 218 (1997), 53-68 · Zbl 0917.11035
[3]	BERNIK V. I.-DODSON M. M.: Metric Diophantine Approximation on Manifolds. Cambridge Tracts in Math. 137, Cambridge Univ. Press, Cambridge, 1999. · Zbl 0933.11040
[4]	BERNIK V.-KLEINBOCK D.-MARGULIES G. A.: Khintchine type theorems on manifolds: the convergence case for standard and multiplicative versions. Int. Math. Res. Not. 9 (2001), 453-486. · Zbl 0986.11053 · doi:10.1155/S1073792801000241
[5]	BUGEAUD Y.: Approximation by Algebraic Numbers. Cambridge Tracts in Math. 169, Cambridge Univ. Press, Cambridge, 2004. · Zbl 1055.11002
[6]	HARMAN G.: Metric Number Theory. London Math. Soc. Monogr. (N.S.) 18, Claredon Press, Oxford, 1998. · Zbl 1081.11057
[7]	KALOSHA N. I.: Simultaneous approximation of zero values of integral polynomials in \({\mathbb R} \times {\mathbb Q}_p\). Dokl. Nats. Akad. Nauk Belarusi 47 (2003), 19-22. · Zbl 1204.11119
[8]	KHINTCHINE A.: Einige Sätze über Kettenbrüche mit Anwendungen auf die Theorie der Diophantischen Approzimationen. Math. Ann. 92 (1924), 115-125. · JFM 50.0125.01
[9]	KLEINBOCK D.-MARGULIES G. A.: Flows on homogeneous space and Diophantine approximation on manifolds. Ann. of Math. (2) 148 (1998), 339-360. · Zbl 0922.11061
[10]	KLEINBOCK D.-TOMANOV G.: Flows on S-arithmetic homogeneous space andapplication to metric Diophantine approximation. Preprint, Max Plank Institute, Bonn 2003-65 (2003).
[11]	KOVALEVSKAYA E. I.: A metric theorem on the exact order of approximation of zero by values of integral polynomials in \({\mathbb Q}_p\). Dokl. Nats. Akad. Nauk Belarusi 42 (1999), 34-36. · Zbl 1038.11043
[12]	KOVALEVSKAYA E. I.: On the exact order of approximation to zero by values of integral polynomials in \({\mathbb Q}_p\). Preprint, Institute Math. National Academy Sciences Belarus, Minsk 8 (547) (1998).
[13]	SCHMIDT W. M.: Diophantine Approximation. Lecture Notes in Math. 785, Springer-Verlag, New York, 1980. · Zbl 0421.10019
[14]	SPRINDŽUK V. G.: Mahler’s Problem in Metric Number Theory. Nauka i Tehnika, Minsk, 1967 [Transl. Math. Monogr. 25, Amer. Math. Soc, Providence, R.I., 1969].
[15]	SPRINDŽUK V. G.: Metric Theory of Diophantine Approximation. Izdatel’stvo Nauka, Moskva, 1977 [John Wiley, New York-Toronto-London, 1979 (Transl. and by R. A. Silverman)].
[16]	ŽELUDEVICH F.: Simultine diophantishe Approximationen abhängiger Grössen in mehreren Metriken. Acta Arith. 46 (1986), 285-296.